Notation for Censored Quantile Regression

Let T be a dependent variable, such as a survival time, and let x be a p1 covariate vector. Quantile regression methods focus on modeling the conditional quantile function, , which is defined as

For example, is the conditional median quantile, and is the conditional quantile function that corresponds to the 95th percentile.

A linear quantile regression model for has the form . One of the advantages of quantile regression analysis is that the covariate effect can change with . Unlike ordinary least squares regression, which estimates the conditional expectation function , quantile regression offers the flexibility to model the entire conditional distribution.

Given observations , standard quantile regression estimates the regression coefficients by minimizing the following objective function over b:

where

However, in many applications, the responses are subject to censoring. For example, in a biomedical study, censoring occurs when patients withdraw from the study or die
from a cause that is unrelated to the disease being studied.

Let denote the censoring variable. In the case of right-censoring, the triples are observed, where and are the observed response variable and the censoring indicator, respectively. Standard quantile regression can lead to a
biased estimator of the regression parameters when censoring occurs.

The following sections describe two methods for estimating the quantile coefficient in the presence of right-censoring.